The bell you are standing on

On 26 December 2004 an earthquake off Sumatra rang the whole planet, and it kept ringing for weeks. This page takes eight days of that ringing, and eight days of the Tohoku earthquake’s ringing six years later, from every seismometer in the global network that returned an unbroken record, and speeds the clock up by a factor of 262,144 so the notes land in your ears. Then it does the same to eight days when nothing happened, so you can hear what an unstruck planet sounds like.

What you are listening to

The ringing lasts far longer than the eight days you just heard. The breathing mode 0S0 has a published attenuation corresponding to a quality factor of about 5714, which means its amplitude takes about 26 days to fall by a factor of e. Weeks, not hours, is the honest word for how long a great earthquake leaves the planet sounding.

Nothing in that sound was synthesised, filtered or tuned. It is the number the seismometer wrote down, once every ten seconds, for eight days: 69,120 integers. The only thing changed is the clock. Playing them at 26214.4 samples a second instead of one every ten seconds multiplies every frequency in the record by exactly 262,144, which is 18 octaves, because 2¹⁸ is a whole number of doublings and a doubling is an octave. Eight days becomes 2.637 seconds, and every interval inside the sound is exactly the interval that was in the ground. A ratio does not care what units you keep time in.

So the pitch you hear is real in the only sense that matters: two tones an octave apart in your speakers were an octave apart underground. What is not real is the loudness. Each recording is scaled to fill the same range, because otherwise the quiet window would be silence, and silence proves nothing about its shape. The largest excursion in each record, in raw instrument counts, is printed next to it, so the true ratio is on the page and not hidden in a gain knob.

The obvious question, then. The earthquake is a hammer blow. Does the sound you just heard belong to the hammer or to the bell?

The same question, drawn

Take those eight days and ask which frequencies are in them. Do it for every station in the global network that returned an unbroken record, and average the answers, so that no single quiet valley or noisy island decides what the planet sounds like. The same arithmetic, in the same order, for all four windows: two earthquakes and two stretches of ordinary Earth.

One vertical scale for all four curves, so their heights are comparable. Read the spikes, not the smooth parts: each station was divided by its own energy across the band before averaging, so a window whose energy is concentrated in lines has its continuum pushed down relative to one whose energy is spread out. The quiet curve sitting above the earthquake’s baseline is that effect, not a quieter earthquake.

The two earthquake windows are a comb. The two quiet windows are a hill with no teeth in it. Whatever is making those lines is switched on by a great earthquake and is not there otherwise, which is the first half of the answer.

The second half is that the two combs are the same comb, and the two earthquakes had nothing in common except the planet they happened on. Their epicentres are 6097 kilometres apart, which is 54.8 degrees of arc, and six years separates them.

Counting it, rather than looking at it

A picture can be talked into anything, so here is the same thing as a count. A line is defined before any of the four spectra is looked at, and identically for all of them: a local maximum standing at least 4 times the median height of the neighbourhood around it, with anything within 4 microhertz of a taller line discarded as the same bump. Then the counter is run.

Then the test. Take the 36 lines from 2004 and ask how many of them have a 2011 line within 2 microhertz. The answer is 26. To know whether that is surprising, the same counter is run against 2000 sets of the same number of frequencies scattered at random across the same band: those score 1.05 on average, and their best score in 2000 tries is 5.

The lines that did not match are worth a sentence, because they are not failures. An earthquake excites the modes its own geometry happens to be good at exciting, so two different sources light up overlapping but not identical sets. Every line in the 2004 comb that is missing from the 2011 comb is a mode the second earthquake was worse at ringing, not a mode that moved.

The one note nobody can disagree about

Look closely at the lines the two earthquakes share and something odd shows up. Most of them agree to within a few hundred nanohertz, which is already good. Three of them agree to within about ten nanohertz, which is a hundred times better. And the three are not a random three: they are exactly the modes in which the Earth is moving straight in and out, the whole sphere breathing, with no pattern around the surface at all.

The reason is a piece of bookkeeping. A mode with a pattern of angular order l around the sphere is really 2l+1 modes with almost the same frequency, and the Earth’s rotation pulls them apart, the way a magnetic field pulls apart a spectral line. Which of them you see depends on where the earthquake was and where you are standing, so two different earthquakes hand you slightly different mixtures and the peak sits in a slightly different place.

The breathing modes have l = 0. That makes 2l+1 = 1. There is nothing to split, nothing to mix, and no way for the source or the station to shade the answer. Everyone gets the same number.

The obvious objection is that those three might simply be the loudest lines, and loud lines are easier to locate. So the test below is a rank test, which ignores how big the numbers are, and it is worked out exactly by counting every one of the 1,540 ways three of the 22 identified lines could have been chosen. The three radial modes rank second, third and fourth smallest. One split multiplet, 0S12, beat all three, which is what a distribution with a median near half a microhertz does occasionally.

The gravest of these unsplittable modes, 0S0, is the whole planet swelling and shrinking once every 20 minutes 28 seconds. It is the sound in the first player that you cannot quite pick out, because at eighteen octaves up it lands near 213.6 Hz, in the middle of everything else. Click its line in the spectrum above to hear it alone.

The lowest note of all, which this page cannot quite claim

The gravest thing the Earth does is not the breathing mode. It is 0S2, a football-shaped wobble with a published frequency of 0.30945 millihertz, one full cycle every 53 minutes 52 seconds. It sits where a seismometer’s own noise floor is climbing steeply, and in these stacks it does not reach the line threshold of 4, so it is not in any count above and this page does not claim it as a detection.

It is visible, though, and what it looks like is the whole splitting argument made concrete. Counting local maxima above 1.8 times the local floor rather than 4, the quiet windows show nothing at all there, and each earthquake window shows not one hump but several, spread across about twenty microhertz. That is what a five-singlet multiplet, pulled apart by the rotation of the Earth and sampled differently by two different earthquakes, is supposed to look like.

Is the number right?

These modes have been measured since the Chilean earthquake of 1960, and the reference compilation for them is the Reference Earth Model tables kept at Scripps. Comparing this page’s numbers to those is the honest test, and it comes with a trap that is worth walking into deliberately.

The trap: across 77 stations the readings of 0S0 scatter by only 23 nanohertz, taken as a median absolute deviation so that two stations whose peak was somewhere else entirely cannot set the number. The standard error of the median that follows is about 3 nanohertz. That figure is real and it is also useless, because it measures only the part of the error that differs between stations. Every station here was handed the same window length, the same taper and the same delay after the earthquake. If those choices push the answer, they push all 77 the same way and the scatter never notices.

So the choices were varied instead. Sixteen stations were re-fetched with 30-day records and the same measurement redone at five window lengths, four delays after the earthquake and three tapers, which is 51 different reasonable ways of asking the same question.

Read that table honestly and it says two things. The first is that this page’s six radial-mode frequencies agree with the published ones to within about 411 nanohertz, which is roughly what an eight-day window and a peak-picker can be expected to deliver, and that for every mode with two independent published values those two disagree with each other by more than their own quoted errors, in one case by 1.82 microhertz. At this precision the literature is not a single number; it is a small cloud.

The second thing is less comfortable. 0S0 is the one radial mode in that table with only a single published value, and this page sits 0.27 microhertz above it, which is slightly more than the 217 nanohertz that varying every processing choice produced. So something small is unaccounted for. Whitening the background before picking the peak moves it by three nanohertz, deconvolving the instrument moves it by none, both earthquakes give the same answer, and 30-day records give the same answer as eight-day ones, so it is none of those. It could be a bias my sweep does not reach, or it could be that the reference value’s stated error of 0.01 microhertz is an internal precision rather than an agreement with a different method. This page cannot tell from the inside, and says so rather than rounding until the numbers touch.

Which raises the obvious objection to the previous section: if my absolute accuracy is only a few hundred nanohertz, how can two earthquakes agree to three? Because those are different quantities. The two windows went through byte-identical code, so whatever bias the window length and the taper introduce is the same bias in both, and it subtracts out of the difference. A comparison can be far sharper than either of the things being compared. That is not a loophole; it is why differential measurements are worth making, and it is why the three-nanohertz agreement is a statement about the Earth and the few-hundred-nanohertz spread is a statement about me.

Three ways this could have been wrong

It could have been the instrument

Everything above runs on raw instrument counts, which are not ground motion. A seismometer’s sensitivity falls away steeply below its corner period, so the overall shape of these spectra is partly the instrument’s shape rather than the Earth’s. That is why this page claims nothing whatever about the heights of the lines. It claims where they are.

A response can move a line only if it has structure as narrow as the line, and it does not; it is smooth over hundreds of microhertz. But an argument is not a measurement, so the measurement was made: the same eight days were fetched twice from the same station, once raw and once with the full instrument response deconvolved to ground displacement, and both went through the identical pipeline.

The quiet windows could have been cherry-picked

They were chosen by a rule written before the data was fetched, and the rule is re-checked against the United States Geological Survey catalogue every time the code runs: no earthquake of moment magnitude 7.5 or greater inside the window, and none in the twenty-one days before it. Both candidates passed with zero events in either interval. The exact queries are in the repository and in windows.json.

The peaks could be an artefact of the arithmetic

Zero-padding a spectrum does not create lines; it interpolates between the ones the data already implies. The proof that the pipeline is not manufacturing structure is sitting in the plot: the quiet windows went through exactly the same padding, the same taper and the same stacking, and came out with 0 lines between them.

The apparatus

Where the numbers come from. Vertical very-long-period channels (VHZ, one sample every 10 seconds) from the Global Seismographic Network, networks IU and II, requested one station at a time from EarthScope’s public web service. Across the two earthquake stacks that is 102 distinct stations, from -89.9 to 82.5 degrees of latitude: South Pole Remote Earth Science Observatory (Quiet Zone) to Alert, NU, Canada. Every request URL is stored next to the file it produced. Windows are 8.000 days, opening five hours after each earthquake so the record is a standing pattern rather than the first passes of the surface waves.

What was done to them. First continuous segment only; the least-squares straight line removed; a Hann taper; zero-padded to 2²⁰ samples and Fourier transformed; each station divided by its own root-mean-square across the band before averaging, so that a single loud station cannot shape the stack. A station is dropped only for having a gap in the first eight days, a wrong sample rate, or a dead trace, and the number dropped is in the table above beside the number kept.

What is not claimed. Not the amplitudes. Not the widths, which would need the attenuation of each mode and a longer record. Not the individual split components, which an eight-day window cannot resolve. Not the modes below about 0.3 millihertz, where the noise floor of a seismometer climbs steeply and this stack has nothing to say. The curve drawn in the plot is thinned by keeping the maximum of every group of four frequency bins, so it is honest about peak heights and good to about 190 nanohertz in position; every measured number on this page comes from the unthinned arrays, never from the picture.

Reproducing it. The seven programs that fetched, stacked, counted and checked all of this are in research/earth-free-oscillations/ in this project’s repository, numbered in the order they run, along with the reference tables and a verifier that recomputes every number on this page from the raw counts and fails if any of them has drifted.

Sources. Waveforms: EarthScope Consortium (formerly IRIS) Data Services, service.iris.edu. Earthquake catalogue: USGS FDSN event service. Reference mode frequencies: the REM (Reference Earth Model) mode tables, subgroup coordinated by Guy Masters, page maintained by Gabi Laske, Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, igppweb.ucsd.edu/~gabi/rem.dir/surface/. The radial-mode column used here is labelled by that site as Masters, singlet stripping, unpublished; the spheroidal values are its reference compilation drawn from the published studies it lists.